| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrniotaidvalN | Structured version Visualization version GIF version | ||
| Description: Value of the unique translation specified by identity value. (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| ltrniotaidval.b | ⊢ 𝐵 = (Base‘𝐾) |
| ltrniotaidval.l | ⊢ ≤ = (le‘𝐾) |
| ltrniotaidval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| ltrniotaidval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| ltrniotaidval.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
| ltrniotaidval.f | ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) |
| Ref | Expression |
|---|---|
| ltrniotaidvalN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 = ( I ↾ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ltrniotaidval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 2 | ltrniotaidval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 3 | ltrniotaidval.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 4 | ltrniotaidval.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
| 5 | ltrniotaidval.f | . . . 4 ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) | |
| 6 | 1, 2, 3, 4, 5 | ltrniotaval 41206 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑃) |
| 7 | 6 | 3anidm23 1441 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑃) |
| 8 | simpl 486 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 9 | 1, 2, 3, 4, 5 | ltrniotacl 41204 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
| 10 | 9 | 3anidm23 1441 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
| 11 | simpr 488 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
| 12 | ltrniotaidval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 13 | 12, 1, 2, 3, 4 | ltrnideq 40800 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹 = ( I ↾ 𝐵) ↔ (𝐹‘𝑃) = 𝑃)) |
| 14 | 8, 10, 11, 13 | syl3anc 1391 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹 = ( I ↾ 𝐵) ↔ (𝐹‘𝑃) = 𝑃)) |
| 15 | 7, 14 | mpbird 259 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 = ( I ↾ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1561 ∈ wcel 2143 class class class wbr 5101 I cid 5542 ↾ cres 5650 ‘cfv 6522 ℩crio 7353 Basecbs 17246 lecple 17294 Atomscatm 39888 HLchlt 39975 LHypclh 40609 LTrncltrn 40726 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-riotaBAD 39578 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-iin 4953 df-br 5102 df-opab 5164 df-mpt 5183 df-id 5543 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-1st 7971 df-2nd 7972 df-undef 8254 df-map 8811 df-proset 18327 df-poset 18346 df-plt 18361 df-lub 18377 df-glb 18378 df-join 18379 df-meet 18380 df-p0 18456 df-p1 18457 df-lat 18465 df-clat 18532 df-oposet 39801 df-ol 39803 df-oml 39804 df-covers 39891 df-ats 39892 df-atl 39923 df-cvlat 39947 df-hlat 39976 df-llines 40123 df-lplanes 40124 df-lvols 40125 df-lines 40126 df-psubsp 40128 df-pmap 40129 df-padd 40421 df-lhyp 40613 df-laut 40614 df-ldil 40729 df-ltrn 40730 df-trl 40784 |
| This theorem is referenced by: dihpN 41961 |
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